Evaluation of the Sparsity in Networks of Sigma-Pi Units when

4.3 Using Networks of Sigma-Pi Units for the Learning and Query of Redundant Mappings, and for Robot Control

θ2

l

θ3

θ4

θ1

Figure 4.13: Schematic drawing of the kinematic chain used for the analyzes described in the text. The chain is composed ofnsegments, withn∈2,3,4, ending with either the blue, green or red segment, respectively. All segments are of equal lengthl, with a total length of the chain of 1.

network weights are arranged in a hyper-cube instead of a grid, and the line of knots corresponds to a slice through the hyper-cube.

4.3.2 Evaluation of the Sparsity in Networks of Sigma-Pi Units when

0 1 23 4 5 10 20 n= 2 n= 3 n= 4

10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

t

(a) Total number of possible synaptic connec-tions (solid lines) and actual number of non-zero weights after training (shown both as dashed lines, and as colored regions below solid lines).

0 12 3 4 5 10 20

n= 2 n= 3 n= 4

10⁻² 10⁻¹ 10⁰

t

(b) Ratio between the number of non-zero weights after training and the total number of possible synaptic connections.

Figure 4.14: Comparison between the total number of possible synaptic connections in a network of sigma-pi units, and the number of non-zero weights after training. Blue, green and red graphs correspond to the end-point of the second, third and fourth segment, respectively (cf. Figure 4.13). Solid lines in (a) show the total number of possible synaptic connections, dotted lines show the number of non-zero weights after training. Colored regions under the solid lines also correspond to the percentage of non-zero weights, to visualize the relationship between zero and non-zero weights in the trained networks.

neurons due to a too large kernel size, but instead would activate either 2ⁿ neurons, or only one, if the input coincided exactly with a receptive field center. The mathematical details for the transformation into barycentric coordinates are given in Appendix A.

To train the connection weights of the sigma-pi units, the n-dimensional space of configurations θ was sampled, input units activated and connection weights updated according to Equation 4.20. The posture space was sampled for 30 values along each dimension, thus yielding training sets of 30ⁿsamples (900, 27.000 and 810.000 samples for 2, 3 and 4 joints, respectively).

Figure 4.14(a) shows for all combinations of n and t the total number of possible synaptic connections, and the proportions of zero and non-zero weights after training.

It can be seen that while the number of non-zero weights does increase exponentially with the number of input dimensions, it only takes up a fractional amount of the total number of possible connections, when 10 or more neurons are used to cover each dimension. This can also be seen in Figure 4.14(b), which shows the ratio between the total number of possible connections and the number of non-zero weights.

To see how many neurons are required in every dimension to represent the kinematic function to a sufficient degree of precision, each network was queried for the forward kinematics. Since the forward kinematics is a proper function, mapping each posture θ onto exactly one end-point position x, it was not necessary to select among sets of

4.3 Using Networks of Sigma-Pi Units for the Learning and Query of Redundant Mappings, and for Robot Control

0 2 4 10 20

10⁻³ 10⁻² 10⁻¹ 10⁰

1 3 5

n= 2 n= 3 n= 4

t

Figure 4.15: Mean squared errors obtained with different network sizes and numbers of segments in the kinematic chain. See text for explanation.

solutions. Thus, the responses of the trained networks were directly used to readout estimations of end-point positions for different postures. The posture space was again sampled with 30 values in each dimension. The estimated end-point position for each posture was compared with the true end-point position, and the mean squared error was computed for all samples as

MSEn,t= 1 30ⁿ

30ⁿ

X

k=1

kex_k,t−x_kk², (4.23) wherex_kis the true end-point position for thek-th sample configuration, andex_k,tis its estimate, which was readout from the query of the corresponding network. Figure 4.15 shows the obtained values for the mean squared error for the different combinations ofn andt. It can be seen that the mean squared error behaves similarly for different values of n, starting off high and decreasing exponentially with the number of neurons used in each dimension. With 5 or less neurons, the error remains rather high (above 0.1, which corresponds to 10% of the total length of the kinematic chain). With 10 neurons in each dimension, the error lies between 0.01 and 0.03 for all values of n, which can be deemed acceptable. Note that these results would correspond to open-loop control, while more precise results can be obtained through online adjustment of the posture using sensory feedback, which will be discussed later in Section 4.3.4.

Table 4.1 summarizes the results of the analyzes for all combinations oft andn.

Networks of sigma-pi units represent a straightforward way to implement internal model learning with the possibility to restore redundant solutions. Inputs are trans-formed into population codes and associations learned between co-activated input neu-rons. This allows to train networks for any form of internal model (cf. Section 4.1), by simply representing all inputs and outputs of the internal model as population codes.

However, networks of sigma-pi units suffer from the exponential growth of the number of possible connections with the number of input dimensions. Using a sparse

imple-Total number of possible synaptic connections

Number of non-zero synaptic weights (ratio be-tween non-zero weights and number of possible connections)

t n= 2 n= 3 n= 4 n= 2 n= 3 n= 4

2 16 32 64 16 (1) 32 (1) 64 (1)

3 81 243 729 77 (0.951) 243 (1) 729 (1)

4 256 1.024 4.096 222 (0.867) 958 (0.936) 4.022 (0.982)

5 625 3.125 15.625 390 (0.624) 2.316 (0.741) 12.282 (0.786) 10 10.000 100.000 1.000.000 1.752 (0.175) 20.404 (0.204) 232.656 (0.233) 20 160.000 3.200.000 64.000.000 5.616 (0.035) 131.908 (0.041) 2.976.312 (0.047) Table 4.1: Summary of results for the analyzes of the relationship between the total number of possible synaptic connections and the number of non-zero weights after training in a network of sigma-pi units.

mentation makes the simulation of larger networks more feasible, since the percentage of non-zero weights in a trained network decreases exponentially with the number of neurons used to cover each dimension, as was demonstrated in the last section.

Still, also the number of non-zero weights increases exponentially with the number of input dimensions. To make the network computationally more efficient, changes to the network properties would have to be made, for example by allowing for variable receptive fields, to cover larger regions in high-dimensional spaces where the mapping to be learned is uniform in character, e.g. close to linear. This would render the network more similar to learning methods that are based on the use of many locally-linear models (as described in the beginning of Section 4.3), and would allow to substantially reduce the number of necessary units.

Internal models are represented in networks of sigma-pi units by locally generalizing radial basis functions (Mel and Koch, 1989), meaning that generalization from train-ing samples only happens within the radius of the receptive fields of input neurons.

Therefore, the number of training samples that are necessary to train the network has to increase with the number of input neurons used, since otherwise (if the resolution of input neurons is higher than the resolution of training samples) the mapping would not be represented for spaces in between training samples. In the example application of learning a kinematic transformation, which was used for the analyzes above, a good trade-off between performance and cost would be to use aroundt= 10 neurons for each input dimension, which gives a rather low mean squared error, while not requiring to draw intractably many training samples for an accurate mapping to be learned.